Unit disk graphs are intersection graphs of circles of unit radius in the plane. We present simple and provably good heuristics for a number of classical NP-hard optimization problems on unit disk graphs. The problems considered include maximum independent set, minimum vertex cover, minimum coloring and minimum dominating set. We also present an on-line coloring heuristic which achieves a competitive ratio of 6 for unit disk graphs. Our heuristics do not need a geometric representation of unit disk graphs. Geometric representations are used only in establishing the performance guarantees of the heuristics. Several of our approximation algorithms can be extended to intersection graphs of circles of arbitrary radii in the plane, intersection graphs of regular polygons, and to intersection graphs of higher dimensional regular objects.

Recent work has looked at extending the k-Means algorithm to incorporate background information in the form of instance level must-link and cannot-link constraints. We introduce two ways of specifying additional background information in the form of δ and constraints that operate on all instances but which can be interpreted as conjunctions or disjunctions of instance level constraints and hence are easy to implement. We present complexity results for the feasibility of clustering under each type of constraint individually and several types together. A key finding is that determining whether there is a feasible solution satisfying all constraints is, in general, NP-complete. Thus, an iterative algorithm such as k-Means should not try to find a feasible partitioning at each iteration. This motivates our derivation of a new version of the k-Means algorithm that minimizes the constrained vector quantization error but at each iteration does not attempt to satisfy all constraints. Using standard UCI datasets, we find that using constraints improves accuracy as others have reported, but we also show that our algorithm reduces the number of iterations until convergence. Finally, we illustrate these benefits and our new constraint types on a complex real world object identification problem using the infra-red detector on an Aibo robot.

JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact support@jstor.org.. INFORMS is collaborating with JSTOR to digitize, preserve and extend access to Operations Research.The dispersion problem arises in selecting facilities to maximize some function of the distances between the facilities. The problem also arises in selecting nondominated solutions for multiobjective decision making. It is known to be NP-hard under two objectives: maximizing the minimum distance (MAX-MIN) between any pair of facilities and maximizing the average distance (MAX-AVG). We consider the question of obtaining near-optimal solutions. For MAX-MIN, we show that if the distances do not satisfy the triangle inequality, there is no polynomial-time relative approximation algorithm unless P = NP. When the distances satisfy the triangle inequality, we analyze an efficient heuristic and show that it provides a performance guarantee of two. We also prove that obtaining a performance guarantee of less than two is NP-hard. For MAX-AVG, we analyze an efficient heuristic and show that it provides a performance guarantee of four when the distances satisfy the triangle inequality. We also present a polynomial-time algorithm for the 1-dimensional MAX-AVG dispersion problem. Using that algorithm, we obtain a heuristic which provides an asymptotic performance guarantee of ir/2 for the 2-dimensional MAX-AVG dispersion problem. M any problems in location theory deal with the placement of facilities on a network to minimize some function of the distances between facilities or between facilities and the nodes of the network (Handler and Mirchandani 1979). Such problems model the placement of "desirable" facilities such as warehouses, hospitals, and fire stations. However, there are situations in which facilities are to be located to maximize some function of the distances between pairs of nodes. Such location problems are referred to as dispersion problems (Chandrasekharan and Daughety 1981, Kuby 1987, Erkut and Neuman 1989, 1990, and Erkut 1990) because they model situations in which proximity of facilities is undesirable. One example of such a situation is the distribution of business franchises in a city (Erkut). Other examples of dispersion problems arise in the context of placing "undesirable" (also called obnoxious) facilities, such as nuclear power plants, oil storage tanks, and ammunition dumps (Kuby 1987, Erkut and Neuman 1989, 1990, and Erkut 1990). Such facilities need to be spread out to the greatest possible extent so that an accident at one of the facilities will not damage any of the others. The concept of dispersion is also useful in the context of multiobjective decision making (Steuer 1986). When the number of nondominated solutions is large, a decision maker may be interested i...

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Abstract. We explore the use of instance and cluster-level constraints with agglomerative hierarchical clustering. Though previous work has illustrated the benefits of using constraints for non-hierarchical clustering, their application to hierarchical clustering is not straight-forward for two primary reasons. First, some constraint combinations make the feasibility problem (Does there exist a single feasible solution?) NP-complete. Second, some constraint combinations when used with traditional agglomerative algorithms can cause the dendrogram to stop prematurely in a dead-end solution even though there exist other feasible solutions with a significantly smaller number of clusters. When constraints lead to efficiently solvable feasibility problems and standard agglomerative algorithms do not give rise to dead-end solutions, we empirically illustrate the benefits of using constraints to improve cluster purity and average distortion. Furthermore, we introduce the new γ constraint and use it in conjunction with the triangle inequality to considerably improve the efficiency of agglomerative clustering.

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We study the problem of nding small trees. Classical network design problems are considered with the additional constraint that only a speci ed number k of nodes are required to be connected in the solution. A prototypical example is the kMST problem in which we require a tree of minimum weight spanning at least k nodes in an edge-weighted graph. We show that the kMST problem is NP-hard even for points in the Euclidean plane. We provide approximation algorithms with performance ratio 2 p k for the general edge-weighted case and O(k 1=4 ) for the case of points in the plane.Polynomial-time exact solutions are also presented for the class of decomposable graphs which includes trees, series-parallel graphs, and bounded bandwidth graphs, and for points on the boundary of a convex region in the Euclidean plane.We also investigate the problem of nding short trees, and more generally, that of nding networks with minimum diameter. A simple technique is used to provide a polynomial-time solution for nding k-trees of minimum diameter. We identify easy and hard problems arising in nding short networks using a framework due to T. C. Hu.This paper appeared in a preliminary form as 28].